Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8,5,3,4,7)(9,10,13,16,14,18,12,17)(11,19,22,24,20,23,21,15);(1,3,6,7,5,2,4,8)(9,11)(10,15,18,24)(12,21)(13,22)(14,20)(16,19,17,23);(2,3)(7,8)(10,16)(11,20)(12,13)(15,23)(17,18)(19,24);(1,4,5,6)(2,7,3,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,f^b=fh,e^c=eh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103286);